QFC Quantitative Finance Statistical & Mathematical Analysis 3 — Questions and Answers
Question 1: In the context of principal component analysis (PCA), what does the first principal component represent?
- The variable with the highest mean
- The direction of maximum variance in the data (Correct answer)
- The eigenvector with the smallest eigenvalue
- The variable most correlated with the response
Correct answer: The direction of maximum variance in the data
The first principal component is the linear combination of original variables that captures the maximum variance in the dataset.
Question 2: What is the effect of increasing the number of Monte Carlo paths on the standard error of a price estimate?
- Standard error decreases linearly with N
- Standard error decreases as 1/√N (Correct answer)
- Standard error increases as √N
- Standard error is unaffected by N
Correct answer: Standard error decreases as 1/√N
Monte Carlo standard error equals σ/√N, so quadrupling paths halves the standard error — a slow convergence rate.
Question 3: Which of the following correctly defines the moment generating function (MGF) of a random variable X?
- M_X(t) = E[e^tX] (Correct answer)
- M_X(t) = E[X^t]
- M_X(t) = ln(E[e^X])
- M_X(t) = E[tX²]
Correct answer: M_X(t) = E[e^tX]
The MGF is defined as M_X(t) = E[e^(tX)], and the k-th derivative evaluated at t=0 gives the k-th moment.
Question 4: In a linear regression with p predictors and n observations, how many degrees of freedom do the residuals have?
- n - 1
- p - 1
- n - p - 1 (Correct answer)
- n + p - 1
Correct answer: n - p - 1
Residual degrees of freedom equal n minus the number of estimated parameters (intercept + p slopes), giving n - p - 1.
Question 5: Which interpolation method is commonly used in building yield curves to ensure smooth forward rates?
- Linear interpolation
- Cubic spline interpolation (Correct answer)
- Nearest-neighbor interpolation
- Lagrange polynomial interpolation
Correct answer: Cubic spline interpolation
Cubic spline interpolation ensures continuity of the curve and its first two derivatives, producing smooth forward rate curves.
Question 6: A random variable X follows a log-normal distribution with parameters μ and σ². What is E[X]?
- e^μ
- e^(μ + σ²)
- e^(μ + σ²/2) (Correct answer)
- μ + σ²/2
Correct answer: e^(μ + σ²/2)
If ln(X) ~ N(μ, σ²), then E[X] = e^(μ + σ²/2), which accounts for Jensen's inequality applied to the exponential function.
Question 7: What does the p-value in hypothesis testing represent?
- The probability that the null hypothesis is true
- The probability of obtaining the observed result or more extreme, assuming H₀ is true (Correct answer)
- The power of the test against the alternative
- The probability that the alternative hypothesis is true
Correct answer: The probability of obtaining the observed result or more extreme, assuming H₀ is true
The p-value measures how likely the observed data (or more extreme) would be under the null hypothesis, not the probability that H₀ is true.
QFC Quantitative Finance Statistical & Mathematical Analysis 3
In the context of principal component analysis (PCA), what does the first principal component represent?